19S-Answer.pdf - Numerical Analysis Midterm Exam Spring...

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Numerical Analysis: Midterm Exam Spring 2019 Name: This exam is scheduled for 70 minutes. It is closed book and closed notes. No calculator should be necessary. By signing your name, you agree to the NYU honor code. The exam is worth 38 points. Problem Problem Points Number Points Earned 1 10 2 8 3 6 4 6 5 8 1
MATH-UA:Numerical Analysis (This page intentionally left blank. You can use it for scratch work.)
MATH-UA:Numerical Analysis 1 Simple iteration [2 5 = 10 pts] Consider the function g ( x ) = x 2 - 2 x + 1 . (a) Draw y = g ( x ) and y = x , indicating the two fixed points x 1 x 2 . (b) Determine an exact expression for the fixed points x 1 , x 2 ( hint : use the quadratic formula). (c) Show that x 1 , and x 2 are unstable. i 2 22 2 4 0 2.2 3 1 I a 3 3 T b 2 4 1213 15 12 1213 15 c gilt 22 2 gym I 13 g'Ha It.IT gYHl 1gYsd1 I unstable
MATH-UA:Numerical Analysis (d) Will a simple iteration (fixed point iteration) with a suitably chosen x 0 , where x 0 6 = x 2 , converge to x 2 ? (e) Suggest a different iterative method to determine x 1 and x 2 . d NO e Newton's method
MATH-UA:Numerical Analysis 2 Newton’s method [2 4 = 8 pts] (a) Suppose we want to find a root x of a function f so that f ( x ) = 0 . Write out the Newton iteration to do this, i.e. what is the formula for the sequence of Newton iterates x k .

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